The Kern is the bread and butter run for Los Angeles since it flows through the southernmost part of the Sierra and has great whitewater. It has nearly year around flows due to a very reliable dam releases from Lake Isabella.
This whitewater run starts with occasional Class III and IV rapids for the first five miles. Around mile 5, you should look out for the Class V+ rapid, Royal Flush that is normally portaged by using the portage trail on river right. Royal Flush is a huge rapid with a series of holes and a dangerous undercut wall on the left that current drives into. Hot shot river guides will occasionally run this rapid but it should be taken seriously.
Below here is a great section of Class III and IV rapids including the especially fun Horseshoe Falls. There are a number of take outs and most people will stop just above Democrat Hot Springs.
You will need a Forest Service permit to float this section of whitewater. Permits can be obtained at the Isabella Ranger Station nearby or at the Cannell Ranger Station in Kernville.
Mile 0: Put-in at the river access near Miracle Hot Springs.
Mile 3.3: Dead Man's Curve (IV) is a fun rapid after a big pool.
Mile 4.2: False Flush (III-IV) is a rapid with boulders in the middle just above the Class V+ Royal Flush Rapid.
Mile 4.5: Royal Flush (V+) is big rapid with big holes and obstructions in the river. Most people portage this on the right side.
Mile 8.1: Horseshoe Falls (IV) has three great drops and is considered one of the most difficult rapids on the Lower Kern.
Mile 10.7: Take-out on river left at Democrat Picnic Area.